[blueash]

Flu is a primarily respiratory disease not gastrointestinal. If all of you became ill at the same time, that strongly suggests a common exposure. It could be food borne or exposure to a contagious person. However, food is more likely as it would be uncommon for a contagious person to be able to infect all of you successfully.
Yes you can report this to the health dept. There is NO reason to go to the ER unless you are experiencing a serious illness. Don't use the ER that way. There is a reason the E means emergency.
Your primary care doctor is perfectly capable of ordering stool culture at either an office visit or over the phone. The health department can, but I don't know if they will, also take charge of specimen collection. They will want to know what items, exactly, each person consumed to look for common exposure should a pathogen be detected.
Even if this was a food borne illness, testing may fail to identify the source. You just don't typically test for everything especially if the patient is doing well and there is not a wide outbreak of illness [which is why you started this topic.. good job]